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Consider the matrix > A := ({2, 1 + I, a)|(1 1, 2, 1)|(-1, 1, 1+b1)); 2 1 - I - 1 A:= 1+1 2 1 (4.1) a 1 1+1b (a) Without attempting diagonalization of A, explain why in case of arbitrary real numbers a and b this matrix cannot be diagonalized by a unitary matrix. Find then the values of these real numbers such that A can be diagonalized by a unitary matrix U. Explain your solution. (b) Assuming now the values for a and b found above, find the unitary matrix U and perform the diagonalization of A. NOTE: you can use commands "simplify" and/or "evalf" to complete your calculations if needed. Solution: Consider the matrix > A := ({2, 1 + I, a)|(1 1, 2, 1)|(-1, 1, 1+b1)); 2 1 - I - 1 A:= 1+1 2 1 (4.1) a 1 1+1b (a) Without attempting diagonalization of A, explain why in case of arbitrary real numbers a and b this matrix cannot be diagonalized by a unitary matrix. Find then the values of these real numbers such that A can be diagonalized by a unitary matrix U. Explain your solution. (b) Assuming now the values for a and b found above, find the unitary matrix U and perform the diagonalization of A. NOTE: you can use commands "simplify" and/or "evalf" to complete your calculations if needed. Solution